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Properties

Label 2-975-195.38-c1-0-46
Degree 22
Conductor 975975
Sign 0.972−0.234i0.972 - 0.234i
Analytic cond. 7.785417.78541
Root an. cond. 2.790232.79023
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.597 − 0.597i)2-s + (1.67 − 0.458i)3-s + 1.28i·4-s + (0.724 − 1.27i)6-s + (2.39 + 2.39i)7-s + (1.96 + 1.96i)8-s + (2.58 − 1.53i)9-s − 1.92·11-s + (0.589 + 2.14i)12-s + (−3.00 − 1.98i)13-s + 2.86·14-s − 0.225·16-s + (4.01 + 4.01i)17-s + (0.627 − 2.45i)18-s − 3.56·19-s + ⋯
L(s)  = 1  + (0.422 − 0.422i)2-s + (0.964 − 0.264i)3-s + 0.642i·4-s + (0.295 − 0.519i)6-s + (0.906 + 0.906i)7-s + (0.694 + 0.694i)8-s + (0.860 − 0.510i)9-s − 0.581·11-s + (0.170 + 0.620i)12-s + (−0.834 − 0.551i)13-s + 0.765·14-s − 0.0562·16-s + (0.973 + 0.973i)17-s + (0.147 − 0.578i)18-s − 0.817·19-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((0.972−0.234i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s+1/2) L(s)=((0.972−0.234i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: 0.972−0.234i0.972 - 0.234i
Analytic conductor: 7.785417.78541
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ975(818,⋅)\chi_{975} (818, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 975, ( :1/2), 0.972−0.234i)(2,\ 975,\ (\ :1/2),\ 0.972 - 0.234i)

Particular Values

L(1)L(1) ≈\approx 2.99270+0.356122i2.99270 + 0.356122i
L(12)L(\frac12) ≈\approx 2.99270+0.356122i2.99270 + 0.356122i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−1.67+0.458i)T 1 + (-1.67 + 0.458i)T
5 1 1
13 1+(3.00+1.98i)T 1 + (3.00 + 1.98i)T
good2 1+(−0.597+0.597i)T−2iT2 1 + (-0.597 + 0.597i)T - 2iT^{2}
7 1+(−2.39−2.39i)T+7iT2 1 + (-2.39 - 2.39i)T + 7iT^{2}
11 1+1.92T+11T2 1 + 1.92T + 11T^{2}
17 1+(−4.01−4.01i)T+17iT2 1 + (-4.01 - 4.01i)T + 17iT^{2}
19 1+3.56T+19T2 1 + 3.56T + 19T^{2}
23 1+(−3.88+3.88i)T−23iT2 1 + (-3.88 + 3.88i)T - 23iT^{2}
29 1−2.60T+29T2 1 - 2.60T + 29T^{2}
31 1−8.42iT−31T2 1 - 8.42iT - 31T^{2}
37 1+(5.96+5.96i)T+37iT2 1 + (5.96 + 5.96i)T + 37iT^{2}
41 1+1.16T+41T2 1 + 1.16T + 41T^{2}
43 1+(0.498+0.498i)T+43iT2 1 + (0.498 + 0.498i)T + 43iT^{2}
47 1+(−2.66+2.66i)T−47iT2 1 + (-2.66 + 2.66i)T - 47iT^{2}
53 1+(−4.62+4.62i)T−53iT2 1 + (-4.62 + 4.62i)T - 53iT^{2}
59 1+3.80iT−59T2 1 + 3.80iT - 59T^{2}
61 1+7.40T+61T2 1 + 7.40T + 61T^{2}
67 1+(−7.88−7.88i)T+67iT2 1 + (-7.88 - 7.88i)T + 67iT^{2}
71 1−2.61T+71T2 1 - 2.61T + 71T^{2}
73 1+(−8.89+8.89i)T−73iT2 1 + (-8.89 + 8.89i)T - 73iT^{2}
79 1+7.19iT−79T2 1 + 7.19iT - 79T^{2}
83 1+(11.9+11.9i)T+83iT2 1 + (11.9 + 11.9i)T + 83iT^{2}
89 1−4.47iT−89T2 1 - 4.47iT - 89T^{2}
97 1+(5.19+5.19i)T+97iT2 1 + (5.19 + 5.19i)T + 97iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.20779090505356817024944616367, −8.802312908175207176671226933322, −8.433019187500822413359278277960, −7.76819860725134516487398077742, −6.89529725833466320286467899120, −5.40034353908870283905892609243, −4.65505330887920879185404383065, −3.48185569388774267966891170877, −2.62723449015425296223123472462, −1.82820725537666854815578350834, 1.26178338413845960650176808049, 2.52704761256876043076377380389, 3.93732373929460336732651085670, 4.73534091983892150431398333345, 5.32768155014133889024532733615, 6.81028437018097276100916227903, 7.45574544602501178660516803407, 8.069693038679698357110997818678, 9.291866275234561500412444460228, 9.919558088765467646779138944071

Graph of the ZZ-function along the critical line